On a theorem of Mattila in the p-adic setting

TL;DR

This paper investigates conditions under which the difference set of two subsets in a p-adic modular grid is large for many rotations, using Fourier analysis and restriction theory to establish sharp bounds.

Contribution

It provides sharp density conditions ensuring large difference sets under group actions in the p-adic setting, extending classical results to a new algebraic context.

Findings

Large difference sets occur for a positive proportion of rotations under certain density conditions.

Conditions are sharp up to constant factors in unbalanced cases.

Uses discrete Fourier analysis and restriction/extension theory tools.

Abstract

Let $A, B$ be subsets of $(\mathbb{Z}/p^r\mathbb{Z})^2$. In this note, we provide conditions on the densities of $A$ and $B$ such that $|gA-B|\gg p^{2r}$ for a positive proportion of $g\in SO_2(\mathbb{Z}/p^r\mathbb{Z})$. The conditions are sharp up to constant factors in the unbalanced case, and the proof makes use of tools from discrete Fourier analysis and results in restriction/extension theory.